LONG-TERM YIELD IN AN AFFINE HJM FRAMEWORK ON S+ d FRANCESCA BIAGINI, ALESSANDRO GNOATTO, AND MAXIMILIAN HARTEL Abstract. We develop the HJM framework for forward rates driven by

LONG-TERM YIELD IN AN AFFINE HJM FRAMEWORK ON S+d

FRANCESCA BIAGINI, ALESSANDRO GNOATTO, AND MAXIMILIAN HARTEL

Abstract. We develop the HJM framework for forward rates driven by affineprocesses on the state space of symmetric positive semidefinite matrices. In

this setting we find an explicit representation for the long-term yield in terms

of the model parameters. This generalises the results of [40] and [6], wherethe long-term yield is investigated under no-arbitrage assumptions in a HJM

setting using Brownian motions and Levy processes respectively.

1. Introduction

Long-term interest rates are particularly relevant for the pricing and hedgingof long-term fixed-income securities, pension funds, life and accident insurances,or interest rate swaps with a very long time to maturity. Thus, the modelingof long-term interest rates is the topic of several contributions which however donot provide a unique definition of long-term interest rates or yield. In Section 4we provide a brief discussion on the different conventions concerning the time tomaturity defining the concept of long-term yield in the literature. Several studiesaddress the topic from a more mathematical or a more macroeconomic point of view.The macroeconomic approach is focused on identifying the macroeconomic factorsinfluencing the long-term yield. For example the paper [43] examines the impact ofmonetary and fiscal policies on long-term interest rates and rejects the hypothesisthat long-term interest are overly sensitive to short-term rates. The article [32] alsostudies the impact of macroeconomic news and monetary policy surprises on long-term yields and presents evidence that these factors have significant effects on short-term as well as on long-term interest rates. The work [36] describes a joint modelof macroeconomic and yield curve dynamics where the continuously compoundedspot rate is an affine function dependent on macroeconomic state variables. Withthe help of this model the influence of macroeconomic effects on the long-term yieldcan be measured. The finding of a model that jointly characterises the behaviourof the yield curve and macroeconomic variables as well as state results for theshort-term and long-term interest rates is also the subject of [2] and [19]. In [2] avector autoregression model is used to describe the relationship between interestrates and macroeconomy, whereas [19] uses a latent factor model with the inclusionof macroeconomic variables to model the yield curve. In [41] the yield curve ismodeled by a three-factor model, where the interest rates can be described withthe help of three underlying latent factors which are employed in order to explainthe empirical result of falling long-term yields.

Mathematical approaches consider the long-term yield as an interest rate withtime to maturity tending to infinity. In the textbook [10] as well as in [6], [40], and[54], the long-term yield is defined as the limit of the continuously compounded

Date: September 5, 2016.Key words and phrases. HJM, Affine Process, Long-Term Yield, Yield Curve, Wishart Process.The research leading to these results has received funding from the European Research Council

under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grantagreement no [228087].

1

2 F. BIAGINI, A. GNOATTO, AND M. HARTEL

spot rate. In this paper we adopt this definition. The respective form of the long-term yield then depends on the chosen interest rate model, whereas there can bemade some universal statements concerning the asymptotic behaviour of yields inan arbitrage-free market, independent of the chosen setting.

One of the most important results concerning the asymptotic behaviour of yieldsis that in an arbitrage-free market, long-term zero-coupon rates can never fall, asfirst stated in [22], consequently referred to as DIR-Theorem. This result was maderigorous by [46]. An alternative proof using a different definition of arbitrage can befound in [51]. Then, [37] provided a generalisation of the proof of the DIR-Theorem,where they assume the existence of an equivalent martingale measure and omitsome measurability conditions. The assumption of the existence of an equivalentmartingale measure is relaxed in [39]. Finally, [28] generalised the theorem bydropping the requirement of the existence of the long-term yield, and showed thatthe limit superior of zero-coupon rates and forward rates never fall. From theseresults the general behaviour of the long-term yield has been clarified. Howeveronly a few studies have contributed to find explicitly the form for the long-termyield in specific models.

Since it is very important to investigate the concrete structure of the long-termyield for several applications, our aim here is to provide an explicit representation ofthe long-term yield in an HJM framework driven by a general affine process on S+

d .Concrete computations of the long-term yield as limit of the standard yield havebeen done in [10], [40], [53], [54] in a Brownian motion setting and more recentlyin [6] in a general Levy setting. In [6] an explicit form for the long-term yield isprovided that takes into account the impact of jumps on the long-term behaviour.In this paper our setting presents the main advantage that the forward curve canbe described by taking account of a rich interdependence structure among factors.This provides a flexible way of describing the impact of different risk factors andof their stochastic correlations on the long-term yield. Under some integrabilityand measurability conditions on the parameters, we are able to obtain an explicitform of the long-term yield, which results to be independent of the underlyingprobability measure. This extends a result of Section 2.2 in [40] to a multifactorsetting including jumps. Moreover, we prove that in our context jumps in thedynamics of the yield do not impact the long-term behaviour.

In order to model the long-term yield, we first provide an extension of the clas-sical Heath-Jarrow-Morton framework to a setting where the market is driven bysemimartingale taking values on the cone S+

d of positive semidefinite symmetricd × d matrices. This class of stochastic processes has appealing features and isincreasingly studied in finance research, in particular for modelling multivariatestochastic volatilities in equity and fixed income models, cf. e.g. [5], [16], [17], [18],[30], [47], and [50]. It allows to model a whole family of factors which share non-linear links among each other, providing a more realistic description of the market.In many situations, the presence of stochastic correlations among factors does notcome at the cost of a loss of analytical tractability, as these processes are affine, inthe sense of [13]. The class of affine processes on S+

d , i.e. stochastically continuousMarkov processes with the feature that the Laplace transform can be representedas an exponential-affine function, was introduced to applications in finance by [29]and [30] in the form of Wishart processes, a particular affine process first describedby Bru in [9]. Theoretical background to affine processes on S+

d can be found,among other publications, in [12], [13], [14], [20], [27], [31], and [45]. A first appli-cation of Wishart processes for short rate modelling is given in [26], while a Libormodel using matrix-valued affine processes is constructed in [15]. Here we considerfor the first time an affine HJM framework on S+

d , where we develop formulas for

LONG-TERM YIELD IN AN AFFINE HJM FRAMEWORK ON S+d 3

forward rates, short rates, and continuously compounded spot rates as well as de-termine the HJM condition on the drift. Note also that we allow for general affineprocesses on S+

d , i.e. we admit jumps. This setting provides a flexible and conciseway of taking into account the influence of a large number of factors on interestrates dynamics and represents a further contribution in capturing the dependencestructure affecting the interest rates evolution. Moreover, in the final examples, weshow that in this setting we can originate affine multidimensional realisations forthe forward rate in the sense of [11, Def. 3].

An interesting aspect of our study is the use of a matrix-valued driver, whoseelements are stochastically correlated among each other. Our choice for such richmulti-dimensional dynamics is open for different economic interpretations whichare beyond the scope of the present contribution. Let us remark however that weview our specification as beneficial in two possible contexts. First it provides analternative way to capture the intrinsic multivariate and dynamic nature of the riskfactors’ dependence structure influencing the yield curve. Secondly, if we extendour view to the post-crisis interest rate market, i.e. to a multiple curve interest ratesetting, it can be used in the description of positive spreads among different curvesto take into account the impact of credit and liquidity/systemic risk. Furthermorein the special case of a Wishart process as driving factor we are able to provide thecorrelation structure in a concise way, since the Wishart dynamics automaticallyguarantee that the elements of the driving process are stochastically correlated.

We note that in our setting jumps can be only of finite variation in the multi-dimensional case and the diffusion part can only allow a square-root dependenceon the driving process. However both of these model assumptions are not in theend as restrictive as they may appear. Jumps of infinite variation would producean infinite long term rate as shown by Theorem 3.6 in [6]. Furthermore our settingalso allows for negative interest rates since the forward rate given in (3.1) as wellthe associated short rate in (3.12) are shifted square root processes, which mayturn negative because of the presence of the drift shift. Shifted models are com-monly adopted in the industry in a negative interest rate environment as a tool toaccomodate negative rates while keeping most implementation and computationalaspects unchanged. A prominent example in this sense is the nowadays widespreadadoption of the shifted Black model, see [8], as a quoting mechanism for EuropeanSwaptions by ICAP, see the VCAP page on the Thomson Reuters Eikon platform.The economic intuition behind the adoption of such shifted models is that in anegative interest rate market the probability of an interest rate increase is higherwith respect to a decrease.A further advantage of our framework is that it can be implemented in spite of thefact that we use a relatively complicated state space. There are different imple-mentation strategies for Wishart based interest rate models. Firstly, our examplesshow that we can produce time-inhomogeneous affine models for the short rate. Itis hence possible e.g. to compute bond prices by means of an exponentially affineformula, where the coefficients are solutions of matrix Riccati differential equationswith time-varying coefficients. These solutions can be numerically evaluated. Forthe time-homogeneous case, see [26]. Secondly, we can apply Monte Carlo methodsin conjunction with the discretization of SDEs. In particular, it is possible to dis-cretize the dynamics of the instantaneous forward rate (3.1) or, equivalently, thedynamics of the zero-coupon bond in Proposition 3.1. Such discretizations onlyrequire the availability of sampling schemes for Wishart process which are studiede.g. in [1].Explicit results on asymptotic behaviour of the long term yield are also of wideinterest, especially they could be relevant for the literature on long-term risk, see


in particular the recent contribution [33]. In [33] the goal is to provide a termstructure of risk prices by changing the investment horizon. Our explicit resultscould be then used to provide analytical expression for the growth-rate risk in thelimit.

The paper is structured as follows. In Section 2 we present the main propertiesof affine processes on S+

d as well as features that are important in the course ofthis paper. Then, in Section 3 analytical expressions for different interest rates aredeveloped under the HJM framework with an affine process X on S+

d as stochasticdriver of the forward rate. In Section 4 we provide an explicit representation of thelong-term yield in the HJM framework on S+

d and finally Section 5 considers someconcrete examples.

2. Affine Processes on S+d

Affine processes were initially studied by [21] and later fully characterised by [20]on the state space Rm+ × Rn with m,n ∈ N. The theoretical framework for affine

processes on the state space S+d , can be found in extensive forms in [13] and [44].

In this section, we state, for the reader‘s convenience, the results of these workswhich are used in the course of this paper as well as the basic required notations.In general, for the stochastic background and notation we refer to [49]. Let d ∈ N.Then,Md denotes the set of all d× d matrices with entries in R, Sd is the space ofsymmetric d×d matrices with entries in R, and S+

d stands for the cone of symmetricd× d positive semidefinite matrices with entries in R which induces a partial orderrelation on Sd:

For x, y ∈ Sd it is x y if y − x ∈ S+d .

The space Sd is endowed with the scalar product A ·B := Tr[A>B

]for A,B ∈ Sd,

where Tr [A] denotes the trace of the matrix A, and the norm induced by this scalarproduct.

Throughout this paper, given A ⊆Md, B(A) denotes the Borel σ-algebra on Aand b(A) the Banach space of bounded real-valued Borel-measurable functions fon A with norm ‖f‖∞ = supx∈A |f(x)|.

Let (Ω,F , (Ft)t≥0 ,Px) be a filtered probability space with the filtration (Ft)t≥0

satisfying the usual conditions of completeness and right-continuity and X :=(Xt)t≥0 a stochastic process on this probability space. For x ∈ S+

d , Px is a prob-

ability measure such that Px(X0 = x) = 1. Given t > 0, Xt− := lims↑tXs, wedefine

∆Xt := Xt −Xt− , (2.1)

the jump at t, ∆X0 ≡ 0.Next, we define the transition probabilities for all t ≥ 0 as:

pt : S+d × B

(S+d

)→ [0, 1] , (x,B) 7→ Px(Xt ∈ B) .

Further, let (Pt)t≥0 be a semigroup such that

Ptf(x) :=

∫S+d

f(ξ) pt(x, dξ) = Ex[f(Xt)] , x ∈ S+d , (2.2)

where f ∈ b(S+d

).

We consider a time-homogeneous Markov process X with state space S+d , i.e.

the Markov property holds for all A ∈ B(S+d

), x ∈ S+

d , and s, t ≥ 0 (cf. Definition17.3 in [42]):

Px(Xt+s ∈ A | Fs) = pt(Xs, A) Px - a.s.

Next, we want to define the characteristics of an affine process on S+d (cf. Definition

2.1 in [13]).


Definition 2.1. A Markov process X with values in S+d is called affine if the

following two properties hold:

(i) It is stochastically continuous, i.e. it holds for all t ≥ 0 and all ε > 0:

lims→t

Px(‖Xs −Xt‖ > ε) = 0 .

(ii) Its Laplace transform has exponential-affine dependence on the initial state,i.e. the following equation holds for all t ≥ 0 and u, x ∈ S+

d :

Pte−Tr[ux] (2.2)

=

∫S+d

e−Tr[uξ] pt(x, dξ) = e−φ(t,u)−Tr[ψ(t,u)x] , (2.3)

for some functions φ : R+ × S+d → R+ and ψ : R+ × S+

d → S+d .

From the stochastic continuity of X the weak convergence of the distributionspt(x, ·), t ≥ 0, follows directly, i.e. it holds for all t ≥ 0 (cf. Satz 5.1 in [4]):

lims→t

ps(x, ·) = pt(x, ·) .

Note, that due to the non-negativity of X the Laplace transform is well-definedand can be used to characterise an affine process. Further, in consequence of thestochastic continuity of the process according to Proposition 3.4 in [13], the processX is regular in the sense of Definition 2.2 in [13].

As well we assume that the affine Markov process is conservative, that meansthat the process will remain almost surely on the state space S+

d all the time.

Definition 2.2. The affine process X is called conservative if for all t ≥ 0 thefollowing condition holds:

pt(x, S+

d

)= 1 ,

i.e. Xt ∈ S+d Px-a.s.

Now, we are able to introduce the so-called admissible parameter set whichgeneralises the concept of Levy triplet to the setting of affine processes on S+

d

(cf. Definition 3.1 in [44]).

Definition 2.3. An admissible parameter set (α, b,B,m, µ) consists of

(i) a linear diffusion coefficient α ∈ S+d ,

(ii) a constant drift term b ∈ S+d which satisfies

b (d− 1)α ,

(iii) a Levy measure m on S+d \0 to represent the constant jump term∫

S+d\0

(‖ξ‖ ∧ 1) m(dξ) <∞ , (2.4)

(iv) a linear jump coefficient µ : S+d \ 0 → S+

d \ 0 which is a σ-finite measureand satisfies ∫

S+d\0

(‖ξ‖ ∧ 1) µ(dξ) <∞ , (2.5)

(v) a linear drift B : S+d → S+

d that satisfies the condition

Tr[B(x)u] ≥ 0 for all x, u ∈ S+d with Tr[xu] = 0 .

Theorem 2.1. Suppose X is a conservative affine process on S+d with d ≥ 2.

Then X is regular and has the Feller property. Moreover, there exists an admissible


parameter set (α, b,B,m, µ) such that φ : R+ × S+d → R+ and ψ : R+ × S+

d → S+d

in (2.3) solve the generalised Riccati differential equations for u ∈ S+d

∂tφ(t, u) = F (ψ(t, u)) , φ(0, u) = 0 , (2.6)

∂tψ(t, u) = R(ψ(t, u)) , ψ(0, u) = 0 , (2.7)

with

F (u) := Tr[bu]−∫S+d\0

(e−Tr[uξ] − 1

)m(dξ) , (2.8)

R(u) := −2uαu+B>(u)−∫S+d\0

(e−Tr[uξ] − 1

)µ(dξ) . (2.9)

Conversely, let (α, b,B,m, µ) be an admissible parameter set and d ≥ 2. Then thereexists a unique conservative affine process X on S+

d such that the affine property

(2.3) holds for all t ≥ 0 and u, x ∈ S+d with φ : R+×S+

d → R+ and ψ : R+×S+d →

S+d given by (2.6) and (2.7).

Proof. Cf. Theorem 2.4 of [13] and Theorem 4.1 of [44].

Besides the admissible parameter set, we need to define the matrix variate Brow-nian motion for the representation of the affine process X (cf. Definition 3.23 in[48]).

Definition 2.4. A matrix variate Brownian motion W ∈Md is a matrix consistingof d2 independent, one-dimensional Brownian motions Wij , 1 ≤ i, j ≤ d.

Remark 1. By (3.3) of [44] we obtain that in the case of d ≥ 2, the affine processX has only jumps of finite variation, i.e for all t ≥ 0∫ t

0

∫S+d\0

‖ξ‖ µX(ds, dξ) <∞ . (2.10)

Now, we can state the following representation of X.

Theorem 2.2. Let X be a conservative affine process on S+d , d ≥ 2, with ad-

missible parameter set (α, b,B,m, µ), where Q ∈ Md such that Q>Q = α. Thenthere exists a matrix Brownian motion W ∈ Md such that X admits the followingrepresentation:

Xt = x+

∫ t

0

(b+B(Xs)) ds+

∫ t

0

(√XsdWsQ+Q>dW>s

√Xs

)+

∫ t

0

∫S+d\0

ξ µX(ds, dξ) ,

(2.11)where µX(ds, dξ) is the random measure associated with the jumps of X, having thecompensator

ν(dt, dξ) := (m(dξ) + Tr[Xt µ(dξ)]) dt . (2.12)

Proof. Cf. Theorem 3.4 in [44].

Note, that it is possible to choose Q this way since Q>Q ∈ S+d for all Q ∈ Md

due to Theorem 2.2 (ix) in [48].

Remark 2. If in Theorem 2.2 we have b = δα with δ ≥ 0, B(z) = Mz + zM>

with M ∈Md, and there are no jumps, the process X is a Wishart process, cf. [9].

Throughout this paper we consider X to be a conservative, regular, affine processon the state space S+

d with d ≥ 2, hence X can be represented by equation (2.11).Furthermore, the linear drift coefficient B is of the form

B(z) = Mz + zM> +G(z) , z ∈ S+d , (2.13)


where M ∈ Md and G : Sd → Sd is linear satisfying G(S+d

)⊆ S+

d to encompass awider range of affine processes (cf. (2.30) in [13]).

Note, that in the case of X being not conservative, all subsequent calculationsand the consequential results are still valid, as long as another set of admissibleparameters is used with an additional constant killing rate term c ∈ R+ and anadditional linear killing rate coefficient γ ∈ S+

d . In the case of d = 1, the processX can also have jumps of infinite activity. Therefore, the parameter set has tobe extended by a truncation function for compensating the infinite variation partof the jumps. The most general admissible parameter set, encompassing the caseof X being not conservative on a state space with dimension d = 1, is stated inDefinition 2.3 in [13].

3. Affine HJM Framework on S+d

We now provide a HJM framework to model the forward curve using affineprocesses on S+

d in the setting outlined in Section 2.By a T -maturity zero-coupon bond we mean a contract that guarantees its holder

the payment of one unit of currency at time T , with no intermediate payments. Thecontract value at time t ≤ T is denoted by P (t, T ) and the bond market satisfiesthe following hypotheses: (1) there exists a frictionless market for T -bonds forevery maturity T ≥ 0; (2) P (T, T ) = 1 for every T ≥ 0; (3) for each fixed t, thezero-coupon bond price P (t, T ) is differentiable with respect to the maturity T .

The money market account is βt := exp(∫ t

0rs ds

)with rt denoting the short rate

at time t. We set ∆2 := (t, T ) ∈ R+ × R+, t ≤ T and assume the forward ratesf : Ω×∆2 → R to evolve for every maturity T > 0 according to

f(t, T ) = f(0, T ) +

∫ t

0

α(s, T ) ds+

∫ t

0

Tr[σ(s, T ) dXs] , 0 ≤ t ≤ T, (3.1)

where X is an affine conservative process with representation (2.11) for a giveninitial value x ∈ S+

d . Since we fix the initial value X0 = x, from now on we write Pfor Px. We impose the following conditions on the drift α : Ω×R+ ×R+ → R andthe volatilities σij : Ω× R+ × R+ → R, i, j ∈ 1, . . . , d:1

Assumption 1.

• α := α(ω, s, u) : (Ω×R+×R+,F⊗ B(R+)⊗ B(R+)) → (R,B(R)) is jointlymeasurable.• For all T ≥ 0: ∫ T

0

∫ T

0

|α(s, u)| ds du <∞ P-a.s.

• For all s, u ∈ R+ and a.e. ω ∈ Ω: σ(s, u) ∈ S+d , i.e. σ(s, u) is a symmetric

positive semidefinite d× d matrix.• σij := σij(ω, s, u) : (Ω×R+×R+,F⊗B(R+)⊗B(R+)) → (R,B(R)) are

jointly measurable for all i, j ∈ 1, . . . , d.• For all T ≥ 0: (α(s, T ))s∈[0,T ] and (σ(s, T ))s∈[0,T ] are adapted.

• For all T ≥ 0:

sups,u≤T

‖σ(s, u)‖ <∞ P-a.s.

• For all i, j ∈ 1, . . . , d : σij : R+ × R+ → R is caglad in both components.

1For α and σ we write the shortened version α(s, T ) := α(ω, s, T ) and σ(s, T ) := σ(ω, s, T ).


Due to Assumption 1 the forward rate process is well-defined in (3.1). Note thatother integrability conditions can be chosen to guarantee that the integrals in (3.1)are well-defined. In this case the results of the paper will also apply under technicalmodifications of the proofs.

Proposition 3.1. If X is a conservative affine process and Assumption 1 holds,then for every maturity T > 0 the zero-coupon bond price follows a process of theform

P (t, T ) = P (0, T ) +

∫ t

0

P (s, T ) (rs +A(s, T )) ds

+ 2

∫ t

0

P (s, T ) Tr[Σ(s, T )

√Xs dWsQ

]+

∫ t

0

P (s−, T )

∫S+d\0

(eTr[Σ(s,T ) ξ] − 1

) (µX − ν

)(ds, dξ) , (3.2)

for t ≤ T , where

Σ(s, T ) := −∫ T

s

σ(s, u) du (3.3)

is the T -bond volatility and

A(t, T ) := −∫ T

t

α(t, u) du− F (−Σ(t, T ))− Tr[R(−Σ(t, T ))Xt] , (3.4)

where F and R are given by (2.8), (2.9) respectively.

Proof. The proof can be found in the appendix.

Note that from Assumption 1 it follows that −Σ(t, T ) ∈ S+d for all t, T ≥ 0

since σ(t, T ) ∈ S+d and it is easy to show that

∫ Ttσ(t, u) du ∈ S+

d . Therefore all

necessary integrals are finite with respect to µX , ν, and the compensated jumpmeasure

(µX − ν

), since X has jumps of finite variation and is regular due to

Theorem 2.1. From this it also follows that F (−Σ(t, T )) and R(−Σ(t, T )) exist.

Remark 3. The bond-price process P (t, T ) , 0 ≤ t ≤ T , can be rewritten the fol-lowing way:

P (t, T ) = P (0, T ) +

∫ t

0

P (s, T ) (rs + C(s, T )) ds+

∫ t

0

P (s−, T ) Tr[Σ(s, T ) dXs]

+

∫ t

0

∫S+d\0

P (s−, T )(eTr[Σ(s,T ) ξ]−1−Tr[Σ(s, T ) ξ]

)(µX− ν

)(ds, dξ) ,

(3.5)

with for all 0 ≤ t ≤ T

C(t, T ) := A(t, T )−Tr[Σ(t, T ) (b+B(Xt))]−∫S+d\0

Tr[Σ(t, T ) ξ] (m(dξ) + Tr[Xtµ(dξ)]) ,

(3.6)

where A(t, T ) is defined in (3.4).

Proof. Representation (3.5) follows by (3.2), (2.11), and (3.6).Note that due to Proposition 1.28 of Chapter II in [38] we are able to combine

the measures µX(ds, dξ) and ν(ds, dξ) to(µX− ν

)(ds, dξ), since the affine process

X has only jumps of finite variation (cf. (2.10)) and Assumption 1 guarantees thatall integrals above are finite.

As an immediate consequence of representation (3.2) for the bond price, weobtain the following corollary.


Corollary 3.1. For every maturity T > 0, the discounted zero-coupon bond pricefollows a process of the formula

P (t, T )

βt= P (0, T ) +

∫ t

0

P (s, T )

βsA(s, T ) ds+ 2

∫ t

0

P (s, T )

βsTr[Σ(s, T )

√Xs dWsQ

]+

∫ t

0

P (s−, T )

βs

∫S+d\0

(eTr[Σ(s,T ) ξ] − 1

) (µX − ν

)(ds, dξ) , (3.7)

for all t ≤ T .

Proof. This follows directly from the definition of the money market account andProposition 3.1.

We now investigate the restrictions on the dynamics (3.1) under the assumptionof no arbitrage. Let Q ∼ P be an equivalent probability measure. By Theorem

3.12 of [7] there exists γ ∈ Md with∫ t

0‖γs‖2 ds < ∞ for all t ≥ 0 such that

W ∗t = Wt −∫ t

0γs ds, t ≥ 0, is a matrix variate Brownian motion under Q and an

Ft ⊗ B([0, t])⊗ B(S+d

)measurable function K : Ω× R+ × S+

d \ 0 → R+ with∫ t

0

∫S+d\0

|K(s, ξ)| ν(ds, dξ) <∞ P-a.s.

for all t ≥ 0, such that µX has the Q-compensator

ν∗(dt, dξ) := K(t, ξ) ν(dt, dξ) . (3.8)

Furthermore, for all t ≥ 0

dQdP

∣∣Ft

= Lt

with

logLt =

∫ t

0

γs dWs −∫ t

0

‖γs‖2 ds+

∫ t

0

∫S+d\0

logK(s, ξ)µX(ds, dξ)

+

∫ t

0

∫S+d\0

(1−K(s, ξ)) ν(ds, dξ) . (3.9)

Definition 3.1. Let Q ∼ P. Then Q is an equivalent local martingale measure(ELMM) for the bond market if for all T > 0 the discounted bond price processP (t,T )βt

, t ∈ [0, T ], is a Q-local martingale.

Theorem 3.1 (HJM drift condition on S+d ). A probability measure Q ∼ P with

Radon-Nikodym density (3.9) is an ELMM if and only if

α(t, T ) = −Tr[σ(t, T )

(b+B(Xt) + 2

√Xt γtQ

)]− 4 Tr

[Qσ(t, T )Xt Σ(t, T )Q>

]−∫S+d\0

Tr [σ(t, T ) ξ] eTr[Σ(t,T ) ξ]K(t, ξ) (m(dξ)+Tr[Xsµ(dξ)]) (3.10)

for all T > 0, dt⊗ dP-a.s.In this case, the Q-dynamics of the forward rates f(t, T ) , 0 ≤ t ≤ T , are of the


form

f(t, T ) = f(0, T ) +

∫ t

0

4 Tr

[Qσ(s, T )Xs

∫ T

s

σ(s, u) du Q>]

−∫S+d\0

K(s, ξ) Tr[σ(s, T ) ξ](eTr[Σ(s,T ) ξ]−1

)(m(dξ)+Tr[Xsµ(dξ)])

ds

+

∫ t

0

∫S+d\0

Tr[σ(s, T ) ξ](µX− ν∗

)(ds, dξ)

+ 2

∫ t

0

Tr[σ(s, T )

√Xs dW

∗s Q]. (3.11)

Proof. The proof can be found in the appendix.

Theorem 3.1 shows that the important property of the classical HJM framework,established in [35], that the forward rates are only dependent on the volatility inan arbitrage-free market, still holds in the framework of affine processes on S+

d .Next, we want to investigate how the short rate process rt, t ≥ 0, can be repre-

sented in the current framework.

Corollary 3.2. Suppose that f(0, T ), α(t, T ) and σ(t, T ) are differentiable in T forall t ≥ 0, ∂Tα(t, T ) is jointly measurable, adapted, and caglad in t, and ∂Tσ(t, T )is jointly measurable, adapted, and caglad in t. Further, it holds for all t ≥ 0 that∫ t

0

|∂uf(0, u)| du <∞ ,

as well as ∫R+

∫R+

|∂Tα(t, T )| dt dT <∞ .

Then, the short rate process (rt)t≥0 is of the form

rt = r0 +

∫ t

0

φ(u) du+

∫ t

0

Tr[σ(u, u) dXu] , (3.12)

where

φ(u) := α(u, u) + ∂uf(0, u) +

∫ u

0

∂uα(s, u) ds+

∫ u

0

Tr[∂uσ(s, u) dXs] .

Proof. Representation (3.12) is a consequence of the theorem of Fubini for inte-grable functions (cf. [42], Chapter 14, Theorem 14.16), the stochastic Fubini theo-rem (cf. [49], Chapter IV, Theorem 65) and the characterisation of the short rateprocess that holds for all s ≥ 0

ru := f(u, u)(3.1)= f(0, u) +

∫ u

0

α(s, u) ds+

∫ u

0

Tr[σ(s, u) dXs] . (3.13)

We now calculate the yield process

Y (t, T ) := − logP (t, T )

T − t, 0 ≤ t ≤ T, (3.14)

for T > 0 in the HJM framework for affine processes on S+d . We recall that the

term “yield curve” is used differently in the literature. For example, in [8] it is acombination of simply compounded spot rates for maturities up to one year andannually compounded spot rates for maturities greater than one year. In this paperwe will refer to the function T 7→ Y (t, T ) as yield curve in t, see also Section 2.4.4of [24].


Note that if f : Rn → Sd for some n, d ∈ N, then it is for a, b ∈ R, x ∈ Rn:

Tr

[∫ b

a

f(x)>∂xf(x) dx

]=

1

2

(‖f(b)‖2 − ‖f(a)‖2

). (3.15)

Lemma 3.1. Let 0 ≤ t < T and let X be an affine process as in (2.11). UnderAssumption 1 and the ELMM Q the yield for [t, T ] can be expressed in the compactform

Y (t, T ) = Y (0; t, T ) + 2

∫ t

0

Tr

[Q

Γ(s, T )− Γ(s, t)

T − tQ>]ds

+

∫ t

0

∫S+d\0

eTr[Σ(s,T ) ξ] − eTr[Σ(s,t) ξ]

T − tν∗(ds, dξ)

−∫ t

0

∫S+d\0

Tr [(Σ(s, T )− Σ(s, t)) ξ]

T − tµX(ds, dξ)

− 2

∫ t

0

Tr

[Σ(s, T )− Σ(s, t)

T − t√XsdW

∗sQ

](3.16)

with the continuously compounded forward rate for [t, T ] prevailing at 0 given by

Y (0; t, T ) :=1

T − t

(∫ T

t

f(0, u) du

)(3.17)

and

Γ(s, t) := Σ(s, t)XsΣ(s, t) (3.18)

for all s, t ≥ 0.

Proof. Let 0 ≤ t < T . Note that for 0 ≤ s ≤ t ≤ T it holds

∫ T

t

σ(s, u) du(3.3)= − (Σ(s, T )− Σ(s, t)) . (3.19)

Further, for some a, b, s ≥ 0 it is

∫ b

a

Tr[Qσ(s, u)Xs Σ(s, u) Q>

]du

(3.3)= −

∫ b

a

Tr[Q∂uΣ(s, u)Xs Σ(s, u) Q>

]du

= −∫ b

a

Tr

[(QΣ(s, u)

√Xs

)>∂u

(QΣ(s, u)

√Xs

)]du

(3.15)= −1

2

(∥∥∥QΣ(s, b)√Xs

∥∥∥2

−∥∥∥QΣ(s, a)

√Xs

∥∥∥2)

(3.18)= −1

2Tr[Q (Γ(s, b)− Γ(s, a))Q>

]. (3.20)


Then by applying the Fubini theorems, the yield for [t, T ] is

Y (t,T ) =1

T − t

(∫ T

t

f(t, u) du

)(3.11)

=

∫ T

t

f(0,u)

T − tdu− 4

T − t

∫ T

t

∫ t

0

Tr[Qσ(s, u)XsΣ(s, u)Q>

]ds du

+1

T−t

∫ T

t

∫ t

0

∫S+d\0

Tr[σ(s, u) ξ](µX− ν∗

)(ds, dξ) du

− 1

T−t

∫ T

t

∫ t

0

∫S+d\0

Tr[σ(s, T ) ξ](eTr[Σ(s,T ) ξ] − 1

)ν∗(ds, dξ) du

+2

T−t

∫ T

t

∫ t

0

Tr[σ(s, u)

√Xs dW

∗sQ]du

(3.17)= Y (0; t, T )− 4

T − t

∫ t

0

∫ T

t

Tr[Qσ(s, u)XsΣ(s, u)Q>

]du ds

− 1

T−t

∫ t

0

∫ T

t

∫S+d\0

∂u Tr[Σ(s, u) ξ](µX− ν∗

)(du, dξ) ds

+1

T−t

∫ t

0

∫ T

t

∫S+d\0

∂ueTr[Σ(s,u) ξ] ν∗(du, dξ) ds

− 1

T−t

∫ t

0

∫ T

t

∫S+d\0

∂u Tr[Σ(s, u) ξ] ν∗(du, dξ) ds

+2

T−t

∫ t

0

Tr

[∫ T

t

σ(s, u) du√Xs dW

∗sQ

](3.19)

=(3.20)

Y (0; t, T ) + 2

∫ t

0

Tr

[Q

Γ(s, T )− Γ(s, t)

T − tQ>]ds

+

∫ t

0

∫S+d\0


T − tν∗(ds, dξ)

−∫ t

0

∫S+d\0

Tr [(Σ(s, T )− Σ(s, t)) ξ]

T − tµX(ds, dξ)

− 2

∫ t

0

Tr

[Σ(s, T )− Σ(s, t)

T − t√Xs dW

∗sQ

].

Corollary 3.3. By (2.12), (3.8), and (3.16) we obtain that

Y (t, T ) = Y (0; t, T ) +

∫ t

0

2 Tr

[Q

Γ(s, T )− Γ(s, t)

T − tQ>]

+

∫S+d\0

M(s, t, T, ξ)K(s, ξ)

T − t(m(dξ) + Tr[Xsµ(dξ)])

ds

−∫ t

0

∫S+d\0

Tr [(Σ(s, T )− Σ(s, t)) ξ]

T − t(µX− ν∗

)(ds, dξ)

− 2

∫ t

0

Tr

[Σ(s, T )− Σ(s, t)

T − t√XsdW

∗sQ

](3.21)

with

M(s, t, T, ξ) := eTr[Σ(s,T ) ξ] − eTr[Σ(s,t)ξ] − Tr [(Σ(s, T )− Σ(s, t)) ξ] . (3.22)


Proof. Let 0 ≤ t < T . Then, we have

Y (t, T )(3.16)

= Y (0; t, T ) + 2

∫ t

0

Tr

[Q

Γ(s, T )− Γ(s, t)

T − t Q>]ds

+

∫ t

0

∫S+d\0


T − t ν∗(ds, dξ)

−∫ t

0

∫S+d\0

Tr [(Σ(s, T )− Σ(s, t)) ξ]

T − t µX(ds, dξ)

− 2

∫ t

0

Tr

[Σ(s, T )− Σ(s, t)

T − t√XsdW

∗sQ

]= Y (0; t, T ) + 2

∫ t

0

Tr

[Q

Γ(s, T )− Γ(s, t)

T − t Q>]ds

+

∫ t

0

∫S+d\0

eTr[Σ(s,T ) ξ] − eTr[Σ(s,t) ξ] − Tr [(Σ(s, T )− Σ(s, t)) ξ]

T − t ν∗(ds, dξ)

−∫ t

0

∫S+d\0

Tr [(Σ(s, T )− Σ(s, t)) ξ]

T − t

(µX− ν∗

)(ds, dξ)

− 2

∫ t

0

Tr

[Σ(s, T )− Σ(s, t)

T − t√XsdW

∗sQ

](2.12)

=(3.22)

Y (0; t, T ) +

∫ t

0

2 Tr

[Q

Γ(s, T )− Γ(s, t)

T − t Q>]

+

∫S+d\0

M(s, t, T, ξ)K(s, ξ)

T − t (m(dξ) + Tr[Xsµ(dξ)])

ds

−∫ t

0

∫S+d\0

Tr [(Σ(s, T )− Σ(s, t)) ξ]

T − t

(µX− ν∗

)(ds, dξ)

− 2

∫ t

0

Tr

[Σ(s, T )− Σ(s, t)

T − t√XsdW

∗sQ

].

4. Long-Term Yield in an Affine HJM Setting on S+d

The expression “long-term yield” is subject to different interpretations in theliterature. For instance, the European Central Bank understands the market yieldsof government bonds with time to maturity close to 10 years as long-term interestrates (cf. [23]), whereas in [52] also high-grade bonds with time to maturity longerthan 20 years are examined to investigate long-term yields. In [54] it is pointedout that for the valuation of some financial securities yield curves with maturitiesup to 100 years are necessary. Here we interpret “long-term yield“ as the yieldwith time to maturity going to infinity. This approach, adopted by [6], [10], [22],[40], is useful for modelling interest rates within a long-time horizon because theasymptotic behaviour can give information about the shape of the yield curve in thelong run where only few empirical data is available. Here we study the asymptoticbehaviour of the long-term yield in the affine HJM setting, introduced in Section3.

Throughout this section in the setting outlined in Section 2 we assume directly

that P is an ELMM for P (t,T )βt

, t ∈ [0, T ], for all T > 0. More precisely, X is

a conservative affine process on S+d , d ≥ 2, with representation (2.11), (2.12) on

(Ω,F , (Ft)t≥0 ,P) and the yield takes the form (3.21), where we write ν instead ofν∗ for the sake of simplicity.

Assumption 2. Let Σ(s, t) be defined as in (3.3) for all 0 ≤ s ≤ t and W amatrix variate Brownian motion. There exists a progressively measurable process


w ∈ L(W ) with values in S+d such that for every i, j ∈ 1, . . . , d, wij is a cadlag

process with1√t

∣∣∣Σ(s, t)ij

∣∣∣ ≤ wij(s) P-a.s. (4.1)

for all 0 ≤ s ≤ t and t 6= 0.

Definition 4.1. The long-term yield (`t)t≥0 is the process defined by

`t := limT→∞

Y (t, T ) , (4.2)

where Y (t, T ) , t ∈ [0, T ], is the yield process for T ≥ 0 given by equation (3.14).

Definition 4.2. If the forward rate process is defined as in (3.1), the long-termdrift µ∞(t) , t ≥ 0, is the process on Md given by

µ∞(t) := limT→∞

Γ(t, T )

T − t= limT→∞

Γ(t, T )

TP-a.s. (4.3)

for all t ≥ 0, where Γ(t, T ) , t ∈ [0, T ], is introduced in (3.18) for every T ≥ 0.Furthermore, the long-term volatility σ∞(t) , t ≥ 0, is the process on Md given by

σ∞(t) := limT→∞

Σ(t, T )

T − t= limT→∞

Σ(t, T )

TP-a.s. (4.4)

for all t ≥ 0, where Σ(t, T ) , t ∈ [0, T ], is introduced in (3.3) for every T ≥ 0.

Here we are supposing that the limits (4.2), (4.3) and (4.4) are well-defined. Thelong-term yield can be characterised as an integral of µ∞ and σ∞ by using thefollowing results.

Proposition 4.1. Let 0 ≤ t ≤ T . The long-term yield at 0 is

limT→∞

Y (0; t, T ) = limT→∞

Y (0, T ) = `0 P-a.s.

Proof. Cf. Proposition 3.3 of [6].

Proposition 4.2. Under Assumption 1 and 2, it holds for all t ≥ 0:

limT→∞

2

t∫0

Tr

[Σ(s, T )− Σ(s, t)

T − t√Xs dWsQ

]= 2

t∫0

Tr[σ∞(s)

√Xs dWsQ

], (4.5)

where σ∞(s) , s ≥ 0, is the long-term volatility process defined by equation (4.4),Σ(s, t), s ≥ 0, is defined for all t ≥ 0 as in (3.3), and the convergence in (4.5) isuniform on compacts in probability (ucp).

Proof. Fix t ≥ 0. By Assumption 1 we have that for all compact intervals [a, b]with 0 ≤ a < b

supt∈[a,b]

∣∣∣∣∫ t

0

Tr[2QΣ(s, t)

√Xs dWs

]∣∣∣∣ <∞ P-a.s.

Consequently on every compact interval [a, b]

1

Tsupt∈[a,b]

∫ t

0

Tr[2QΣ(s, t)

√Xs dWs

]T→∞−→ 0 P-a.s.

Therefore1

T

∫ t

0

Tr[2QΣ(s, t)

√Xs dWs

]T→∞−→ 0 in ucp. (4.6)

Next, we define HT := HTs , s ≥ 0, with

HTs := 2

QΣ(s, T )√Xs

T. (4.7)


Then for T →∞ : HTs → 2Qσ∞(s)

√Xs a.s. for all s ≥ 0.

Since we investigate long-term interest rates it is sufficient to impose long times ofmaturity, say T ≥ 1. Due to Assumption 2, we then have that for all 0 ≤ s ≤ Twith T ≥ 1∥∥HT

s

∥∥ (4.7)=

2

T

∥∥∥QΣ(s, T )√Xs

∥∥∥=

2

T

(Tr[√

Xs Σ(s, T )Q>QΣ(s, T )√Xs

])1/2

=2

T

∑i,j,k,l,m,n

√Xij,s Σ(s, T )jkQ

>klQlm Σ(s, T )mn

√Xni,s

1/2

(4.1)

≤ 2

T

∑i,j,k,l,m,n

√T√Xij,s wjk(s) Q>klQlm

√T wmn(s)

√Xni,s

1/2

=2√T

∑i,j,k,l,m,n

√Xij,s wjk(s) Q>klQlm wmn(s)

√Xni,s

1/2

≤ 2

∑i,j,k,l,m,n

√Xij,s wjk(s) Q>klQlm wmn(s)

√Xni,s

1/2

= 2(

Tr[√

Xs w(s) Q>Qw(s)√Xs

])1/2

= 2∥∥∥Qw(s)

√Xs

∥∥∥ =: h(s) .

Further, we have w ∈ L(W ) due to Assumption 2 and we know from Theorem 2.2

that√X ∈ L(W ). By using Theorem 16 in Chapter IV, Section 2 of [49] it follows

h ∈ L(W ). Then, applying the dominated convergence theorem for semimartingales(cf. Theorem 32 in Chapter IV, Section 2 of [49]), we get:∫ t

0

Tr

[2QΣ(s, T )

√Xs

TdWs

]T→∞−→ 2

∫ t

0

Tr[σ∞(s)

√Xs dWsQ

]in ucp. (4.8)

It follows due to Lemma 5.8 of [25], (4.6), and (4.8):

2

∫ t

0

Tr

[Σ(s, T )− Σ(s, t)

T − t√XsdWsQ

]T→∞−→ 2

∫ t

0

Tr[σ∞(s)

√Xs dWsQ

]in ucp.


limT→∞

2

t∫0

Tr

[Q

Γ(s, T )− Γ(s, t)

T − tQ>]ds = 2

t∫0

Tr[Qµ∞(s)Q>

]ds, (4.9)

where µ∞(s) , s ≥ 0, is the long-term drift process defined by equation (4.3), Γ(s, t),s ≥ 0, is defined for all t ≥ 0 as in (3.18), and the convergence in (4.9) is in ucp.

Proof. Fix t ≥ 0. Since the process Γ(s, t) , s, t ≥ 0, is continuous in t for all fixed sand cadlag in s for all fixed t it follows by (4) of Section 2.8 in [3] that Γ(s, t) , s ≥ 0,is bounded on all compact intervals [a, b] with t ∈ [a, b] and 0 ≤ a < b for a.e. ω ∈ Ω,i.e.

supt∈[a,b]

∣∣∣∣∫ t

0

Tr[QΓ(s, t)Q>

]ds

∣∣∣∣ <∞ P-a.s.



1

Tsupt∈[a,b]

∫ t

0

Tr[QΓ(s, t)Q>

]ds

T→∞−→ 0 P-a.s.

Therefore

1

T

∫ t

0

Tr[QΓ(s, t)Q>

]ds

T→∞−→ 0 in ucp. (4.10)

Let us define GT := GTs , s ≥ 0, with

GTs := 2QΓ(s, T )Q>

T. (4.11)

Then for T →∞ : GTs → 2Qµ∞(s)Q> a.s. for all s ≥ 0.By Assumption 2 we have that for all i, j ∈ 1, . . . , d and 0 ≤ s ≤ T :

Γ(s, T )ij(3.18)

= (Σ(s, T )XsΣ(s, T ))ij =∑k,l

Σ(s, T )ikXkl,sΣ(s, T )lj

(4.1)

≤∑k,l

√T wik(s)Xkl,s

√T wlj(s) = T (w(s)Xsw(s))ij . (4.12)

Therefore we have that for all 0 ≤ s ≤ T∥∥GTs ∥∥ (4.11)=

2

T

∥∥QΓ(s, T )Q>∥∥

=2

T

(Tr[QΓ(s, T )Q>QΓ(s, T )Q>

])1/2=

2

T

∑i,j,k,l,m,n

Qij Γ(s, T )jkQ>klQlm Γ(s, T )mnQ

>ni

1/2

(4.12)

≤ 2

T

∑i,j,k,l,m,n

Qij T (w(s)Xsw(s))jkQ>klQlm T (w(s)Xsw(s))mnQ

>ni

1/2

= 2(Tr[Qw(s)Xsw(s)Q>Qw(s)Xsw(s)Q>

])1/2= 2

∥∥Qw(s)Xsw(s)Q>∥∥ =: g(s) ,

where g is a cadlag process. It follows∫ t

0g(s) ds <∞ for all t ≥ 0 by (4) of Section

2.8 in [3] and we can apply the DCT for progressive processes (cf. Corollary 6.26in Chapter 6 of [42]). By using Lemma 5.8 of [25] and (4.10) we obtain (4.9).

Proposition 4.4. Under Assumption 1 and 2, it holds for all t ≥ 0:∫ t

0

∫S+d\0

Tr[(Σ(s, T )−Σ(s, t)) ξ]

T − tµX(ds, dξ)

T→∞−→∫ t

0

∫S+d\0

Tr[σ∞(s) ξ]µX(ds, dξ) ,

(4.13)where σ∞(s) , s ≥ 0, is the long-term volatility process defined by equation (4.4),Σ(s, t), s ≥ 0, is defined for all t ≥ 0 as in (3.3), and the convergence in (4.13) isin ucp.


Proof. Fix t ≥ 0. First, notice that∣∣∣∣∣∫ t

0

∫S+d\0

Tr[Σ(s, t) ξ]µX(ds, dξ)

∣∣∣∣∣ ≤∫ t

0

∫S+d

|Tr[Σ(s, t) ξ]|µX(ds, dξ)

≤√t

∫ t

0

∫S+d\0

1√t‖Σ(s, t)‖ ‖ξ‖µX(ds, dξ)

(4.1)

≤√t

∫ t

0

∫S+d\0

‖w(s)‖ ‖ξ‖µX(ds, dξ)

≤√t supu∈[0,t]

‖w(u)‖∫ t

0

∫S+d\0

‖ξ‖ µX(ds, dξ) .

Define q(t) :=√t supu∈[0,t] ‖w(u)‖Zt with

Zt :=

∫ t

0

∫S+d\0

‖ξ‖ µX(ds, dξ) , t ≥ 0 .

Note that Zt, t ≥ 0, is a well-defined cadlag process by (2.10). Then for all compactintervals [a, b] with 0 ≤ a < b it is

supt∈[a,b]

|q(t)| = supt∈[a,b]

∣∣∣∣∣√t supu∈[0,t]

‖w(u)‖Zt

∣∣∣∣∣ ≤ √b supt∈[a,b]

supu∈[0,t]

‖w(u)‖ supt∈[a,b]

Zt

=√b supu∈[0,b]

‖w(u)‖ supt∈[a,b]

Zt <∞

because of (4) of Section 2.8 in [3] applied for the cadlag processes ‖w(t)‖ , t ≥ 0,and Zt, t ≥ 0. Consequently on every compact interval [a, b]

1

Tsupt∈[a,b]

∫ t

0

∫S+d\0

Tr[Σ(s, t) ξ]µX(ds, dξ)T→∞−→ 0 P-a.s.

Therefore1

T

∫ t

0

∫S+d\0

Tr[Σ(s, t) ξ]µX(ds, dξ)T→∞−→ 0 in ucp. (4.14)

Due to Assumption 2 we have for s, t ≤ T and T ≥ 1 that

|Tr [Σ(s, T ) ξ]|T

≤ 1

T‖Σ(s, T )‖ ‖ξ‖

(4.1)

≤ 1√T‖w(s)‖ ‖ξ‖ ≤ ‖w(s)‖ ‖ξ‖ =: j(s, ξ) .

We first show that the process j is integrable with respect to the random measureµX on [0, t]× S+

d \0 for all t ≥ 0:∫ t

0

∫S+d\0

j(s, ξ)µX(ds, dξ) =

∫ t

0

∫S+d\0

‖w(s)‖ ‖ξ‖ µX(ds, dξ)

≤ supu∈[0,t]

‖w(u)‖Zt <∞

due to (4) of Section 2.8 in [3] applied for the cadlag process ‖w(t)‖ , t ≥ 0, and by(2.10). We have with the DCT and (4.4) that for all fixed t ≥ 0∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]

TµX(ds, dξ)

T→∞−→∫ t

0

∫S+d\0

Tr[σ∞(s) ξ]µX(ds, dξ) P-a.s. (4.15)

Then, by (4.15) applied for t = b and by (4.17) from Lemma 4.1, it follows that

supt∈[a,b]

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]

TµX(ds, dξ)

T→∞−→ supt∈[a,b]

∫ t

0

∫S+d\0

Tr[σ∞(s) ξ]µX(ds, dξ)


P-a.s. and therefore in probability. Hence by [49], page 57

limT→∞

∫ t

0

∫S+d\0

Tr [Σ(s, T ) ξ]

TµX(ds, dξ) =

∫ t

0

∫S+d\0

Tr[σ∞(s) ξ]µX(ds, dξ) in ucp.

(4.16)By using (4.14) as well as (4.16) it follows (4.13).

Lemma 4.1. With Σ(t, T ) , t ≥ 0, defined as in (3.3) and σ∞(t) , t ≥ 0, as in (4.4)it holds for 0 ≤ a < b∣∣∣∣∣ supt∈[a,b]

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]

TµX(ds, dξ)− sup

t∈[a,b]

∫ t

0

∫S+d\0


∣∣∣∣∣≤∫ b

0

∫S+d\0

∣∣∣∣Tr[Σ(s, T ) ξ]

T− Tr[σ∞(s) ξ]

∣∣∣∣µX(ds, dξ) . (4.17)

Proof. Let 0 ≤ a ≤ b. Then, we have

supt∈[a,b]

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]


t∈[a,b]

∫ t

0

∫S+d\0


= supt∈[a,b]

∫ t

0

∫S+d\0

(Tr[Σ(s, T ) ξ]

T− Tr[σ∞(s) ξ] + Tr[σ∞(s) ξ]

)µX(ds, dξ)

− supt∈[a,b]

∫ t

0

∫S+d\0


≤ supt∈[a,b]

∫ t

0

∫S+d\0

(Tr[Σ(s, T ) ξ]


)µX(ds, dξ) . (4.18)

Furthermore

supt∈[a,b]

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]


t∈[a,b]

∫ t

0

∫S+d\0


= supt∈[a,b]

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]

TµX(ds, dξ)

− supt∈[a,b]

∫ t

0

∫S+d\0

(Tr[σ∞(s) ξ]− Tr[Σ(s, T ) ξ]

T+

Tr[Σ(s, T ) ξ]

T

)µX(ds, dξ)

≥ − supt∈[a,b]

∫ t

0

∫S+d\0

(Tr[σ∞(s) ξ]− Tr[Σ(s, T ) ξ]

T

)µX(ds, dξ)

= inft∈[a,b]

∫ t

0

∫S+d\0

(Tr[Σ(s, T ) ξ]


)µX(ds, dξ) . (4.19)

Hence, it follows from (4.18) and (4.19) that∣∣∣∣∣ supt∈[a,b]

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]


t∈[a,b]

∫ t

0

∫S+d\0


∣∣∣∣∣≤

∣∣∣∣∣ supt∈[a,b]

∫ t

0

∫S+d\0

(Tr[Σ(s, T ) ξ]


)µX(ds, dξ)

∣∣∣∣∣∨

∣∣∣∣∣ inft∈[a,b]

∫ t

0

∫S+d\0

(Tr[Σ(s, T ) ξ]


)µX(ds, dξ)

∣∣∣∣∣≤ supt∈[a,b]

∫ t

0

∫S+d\0

∣∣∣∣Tr[Σ(s, T ) ξ]


∣∣∣∣µX(ds, dξ) .



limT→∞

∫ t

0

∫S+d\0


T − tν(ds, dξ) = 0 , (4.20)

where Γ(s, t), s ≥ 0, is defined for all t ≥ 0 as in (3.18), and the convergence in(4.20) is in ucp.

Proof. Fix t ≥ 0. We note that the left-hand side of (4.20) is equal to

limT→∞

1

T

(∫ t

0

∫S+d\0

(1−eTr[Σ(s,t) ξ]

)ν(ds, dξ)−

∫ t

0

∫S+d\0

(1−eTr[Σ(s,T ) ξ]

)ν(ds, dξ)

).

(4.21)

Hence we study the limit (4.21) and we introduce for all u ∈ S+d

F (u) :=

∫S+d\0

(e−Tr[uξ] − 1

)m(dξ) , (4.22)

R(u) :=

∫S+d\0

(e−Tr[uξ] − 1

)µ(dξ) . (4.23)

Then, we can write due to (2.12), (4.22), and (4.23) that∫ t

0

∫S+d\0

(1−eTr[Σ(s,t)ξ]

)ν(ds, dξ) = −

∫ t

0

(F (−Σ(s, t)) + Tr

[R(−Σ(s, t))Xs

])ds.

The process F (−Σ(s, t)) + Tr[R(−Σ(s, t))Xs

], s ∈ [0, t] , is cadlag for all t ≥ 0

since F and R are continuous functions and X is cadlag. Due to this and (4) ofSection 2.8 in [3] we get that for all compact intervals [a, b] with a, b ≥ 0

supt∈[a,b]

∫ t

0

∫S+d\0


)ν(ds, dξ) <∞ P-a.s.


1

Tsupt∈[a,b]

∫ t

0

∫S+d\0


)ν(ds, dξ)

T→∞−→ 0 P-a.s.

Therefore

1

T

∫ t

0

∫S+d\0


)ν(ds, dξ)

T→∞−→ 0 in ucp. (4.24)

Next, we use the inequality

1−eTr[Σ(s,T ) ξ] = 1−e−Tr[−Σ(s,T ) ξ] ≤ 1∧Tr[−Σ(s, T ) ξ](4.1)

≤ 1∧√T Tr[w(s) ξ] (4.25)

which holds for all ξ ∈ S+d and a.e. ω ∈ Ω, to see that for all 0 ≤ s ≤ T with T ≥ 1,

and for a.e. ω ∈ Ω

1− eTr[Σ(s,T ) ξ]

T

(4.25)

≤ 1

T∧ 1√

TTr[w(s) ξ] ≤ 1 ∧ Tr[w(s) ξ] ≤ 1 ∧ ‖w(s)‖ ‖ξ‖ =: i(s, ξ) .

Since we investigate long-term interest rates it is sufficient to impose long times ofmaturity, say T ≥ 1.


Then, we show that the process i is integrable with respect to the random measureν on [0, t]× S+

d :∫ t

0

∫S+d\0

i(s, ξ) ν(ds, dξ) =

∫ t

0

∫S+d\0

i(s, ξ)(1‖w(s)‖≤1 + 1‖w(s)‖>1

)ν(ds, dξ)

=

∫ t

0

∫S+d\0

i(s, ξ)1‖w(s)‖≤1 ν(ds, dξ)

+

∫ t

0

∫S+d\0

i(s, ξ)1‖w(s)‖>1 ν(ds, dξ)

≤∫ t

0

∫S+d\0

1‖w(s)‖≤1 (1 ∧ ‖ξ‖) ν(ds, dξ)

+

∫ t

0

∫S+d\0

1‖w(s)‖>1 ‖w(s)‖ (1 ∧ ‖ξ‖) ν(ds, dξ)

(2.12)

≤∫ t

0

∫S+d\0

(1 ∧ ‖ξ‖) ν(ds, dξ)

+

∫ t

0

‖w(s)‖1‖w(s)‖>1 ds

∫S+d\0

(1 ∧ ‖ξ‖)m(dξ)

+ Tr

[∫ t

0

‖w(s)‖1‖w(s)‖>1Xs ds

∫S+d\0

(1 ∧ ‖ξ‖)µ(dξ)

]

≤∫ t

0

∫S+d\0

(1 ∧ ‖ξ‖) ν(ds, dξ)

+

∫ t

0

‖w(s)‖ ds∫S+d\0

(1 ∧ ‖ξ‖)m(dξ)

+ Tr

[∫ t

0

‖w(s)‖Xs ds∫S+d\0

(1 ∧ ‖ξ‖)µ(dξ)

]

< ∞ P-a.s.

because of (2.4), (2.5), and (4) of Section 2.8 in [3] applied for the cadlag processesX and ‖w(t)‖Xt, t ≥ 0.Then by the DCT we have that for all t ≥ 0∫ t

0

∫S+d\0

eTr[Σ(s,T ) ξ]−1

Tν(ds, dξ)

T→∞−→ 0 P-a.s. (4.26)

With the same argument as in Proposition 4.4, we then obtain that for 0 ≤ a < b

supt∈[a,b]

∫ t

0

∫S+d\0

1−eTr[Σ(s,T ) ξ]

Tν(ds, dξ) ≤

∫ b

0

∫S+d\0


Tν(ds, dξ) ,

since ν is given by (2.12).This converges to 0 by (4.26) applied for t = b. Therefore a.e. ω ∈ Ω

supt∈[a,b]

∫ t

0

∫S+d\0


Tν(ds, dξ)

T→∞−→ 0 ,

i. e. by [49], page 57∫ t

0

∫S+d\0

eTr[Σ(s,T ) ξ]−1

Tν(ds, dξ)

T→∞−→ 0 in ucp. (4.27)

The result (4.20) follows then by (4.21), (4.24), and (4.27).

We are now ready to state the main result of this section.


Theorem 4.1. Under Assumption 1 and 2, the long-term yield is given by

`t = `0 + 2

∫ t

0

Tr[Qµ∞(s)Q>

]ds, t ≥ 0, (4.28)

with Tr[Qµ∞(s)Q>

]≥ 0 for all 0 ≤ s ≤ t if it exists in a finite form.

Proof. By Lemma 3.1, Proposition 4.1, Proposition 4.2, Proposition 4.3, Proposi-tion 4.4, and Proposition 4.5, the long-term yield can be written in the followingway:

`t = `0 + 2

∫ t

0

Tr[Qµ∞(s)Q>

]ds− 2

∫ t

0

Tr[σ∞(s)

√Xs dWsQ

]−∫ t

0

∫S+d\0

Tr[σ∞(s) ξ]µX(ds, dξ) , t ≥ 0,

whereas the convergence is uniformly on compacts in probability. However if 0 <‖σ∞(t)‖ < ∞ for some t ∈ [0, T ], by (4.4) we have Σ(t, T )ij ∈ O(T − t) for all

i, j ∈ 1, . . . , d. Then we get for all t ≥ 0

Tr[Qµ∞(t)Q>

] (4.3)=∑i,j,k

Qij limT→∞

Γ(t, T )jkT − t

Q>ki

(3.18)= lim

T→∞

1

T − t∑

i,j,k,l,m

Qij Σ(t, T )jlXlm,t Σ(t, T )mkQik

=∞ P-a.s.

That is a contradiction to the existence of the long-term yield. Hence representation(4.28) holds if the long-term yield exists in a finite form. Moreover, we have for allt ≥ 0

Tr[Qµ∞(t)Q>

] (4.3)= lim

T→∞

1

T − tTr[QΓ(t, T )Q>

](3.18)

= limT→∞

1

T − tTr[QΣ(t, T )Xt Σ(t, T )Q>

]= lim

T→∞

1

T − t

∥∥∥√Xt Σ(t, T )Q>∥∥∥2

≥ 0 P-a.s. (4.29)

This concludes the proof.

It follows immediately that (`t)t≥0 is a non-decreasing process. This is a well-known result, which was shown for the first time in 1996 by Dybvig, Ingersoll andRoss in [22] and generally proven in [37]. Our main contribution is to computeexplicitly the form of the long-term yield in dependence of the model’s parameters.In particular, we obtain that the drift µ∞ is given by a stochastic process whichdepends on (the limit of) the volatility and on X. This implies that the representa-tion (4.28) of ` remains the same under a change of equivalent probability measures.This result can be proven by applying the convergence results of Propositions 4.4and 4.5 to the yield expressed in the form (3.21), which provides the form of Yunder a change of equivalent probability measures. In this way we extend a resultof [40] and [6] to a multifactor setting.

To conclude we now discuss some conditions on the volatility process σ(t, T ) thatguarantee the existence of the long-term drift µ∞.

Proposition 4.6. Let σ(t, T ) ∈ O(

1√T−t

)for every t ≥ 0, i.e. σ(t, T )ij ∈

O(

1√T−t

)for all i, j ∈ 1, . . . , d P-a.s. Under the setting outlined in Section


3, we get

Tr[Qµ∞(t)Q>

]<∞ P-a.s.

Proof. Let t ≥ 0 and σ(t, T ) ∈ O(

1√T−t

). Then

Tr[Qµ∞(t)Q>

] (4.29)= lim

T→∞

1

T−t

∥∥∥√Xt Σ(t, T )Q>∥∥∥2

(3.3)= lim

T→∞

1

T−t∑

i,j,k,l,m

Qij

T∫t

σ(t, u)jkdu Xkl,t

T∫t

σ(t, u)lmdu Q>mi

< ∞ P-a.s.

Proposition 4.7. Let σ(t, T ) ∈ O(

1T−t

)for every t ≥ 0 P-a.s. Under the setting

outlined in Section 3, we get

µ∞(t) = 0

and therefore (`t)t≥0 is constant.

Proof. Let t ≥ 0 and σ(t, T ) ∈ O(

1T−t

), i.e. for all i, j ∈ 1, . . . , d it is σ(t, T )ij ∈

O(

1T−t

). Then, we get for all i, j ∈ 1, . . . , d that Σ(t, T )ij ∈ O(log(T − t)) and

therefore for all i, j, k, l ∈ 1, . . . , d that

limT→∞

1

T − tΣ(t, T )ij Σ(t, T )kl = 0 P-a.s. (4.30)

Hence, for all i, j ∈ 1, . . . , d it is

µ∞(t)ij(4.3)= lim

T→∞

Γ(t, T )ijT − t

(3.18)= lim

T→∞

1

T−t∑k,l

Σ(t, T )ikXkl,tΣ(t, T )lj(4.30)

= 0 P-a.s.

By (4.28) this yields `t = `0 for all t ≥ 0, i.e. (`t)t≥0 is constant.

The following table summarises the results regarding the convergence behaviourof the long-term yield for all t ≥ 0.

Long-term drift Long-term volatility Long-term yield Volatility curve

Tr[Qµ∞(t)Q>

]=∞ 0 < ‖σ∞(t)‖ <∞ infinite σ(t, T ) ∼ O(1)

Tr[Qµ∞(t)Q>

]=∞ 0 < ‖σ∞(t)‖ <∞ infinite σ(t, T ) ∼ O(T−t)

Tr[Qµ∞(t)Q>

]= 0 ‖σ∞(t)‖ = 0 constant σ(t, T ) ∼ O

(1

T−t

)0<Tr

[Qµ∞(t)Q>

]<∞ ‖σ∞(t)‖ = 0 non-decreasing σ(t, T ) ∼ O

(1√T−t

)5. Examples

In this section we present two examples for the long-term yield driven by an affineprocess on S+

d under two different volatility specifications. Detailed calculations of

this example can be found in [34]. Consider the affine process on S+d defined as

dXt := aIddt+√XtdWt + dW>t

√Xt (5.1)

for t ≥ 0, a ≥ (d− 1), W a d-dimensional matrix Brownian motion, and Id denot-ing the d-dimensional unit matrix. Here we model directly under the equivalentmartingale measure Q, i.e. we assume ν = ν∗ = 0 and W = W ∗ which impliesγ ≡ 0.


By Theorem 3.1 the Q-dynamics of the forward rate are given by

f(t, T ) = f(0, T )+4

∫ t

0

Tr [σ(s, T )Xs (−Σ(s, T ))] ds+2

∫ t

0

Tr[σ(s, T )

√Xs dWs

]. (5.2)

In the sequel we use the following lemma.

Lemma 5.1. Let (Xt)t≥0 be defined as in (5.1), σ as in Assumption 1 and f ∈C1(R). Then, it holds for all 0 ≤ t ≤ T :

2

∫ t

0

f(s) Tr[σ(s, T )

√Xs dWs

]= −

∫ t

0

f(s) (Tr[Xs ∂sσ(s, T )] + aTr[σ(s, T )]) ds

+ f(t) Tr[σ(t, T )Xt]− f(0) Tr[σ(0, T )X0]

−∫ t

0

f ′(s) Tr[σ(s, T )Xs] ds .

Proof. The proof can be found in [34].

We choose σ(t, T ) as a deterministic symmetric positive semidefinite d×d matrixin the following two different ways

(i)

σ(t, T ) :=

σe−β(T−t) for T ≥ t ,0 for T < t ,

(5.3)

(ii)

σ(t, T ) :=

σ 1√

T−t for T > t ,

0 for T ≤ t .

with σ ∈ S+d , σij ∈ R for all i, j ∈ 1, . . . , d, and β > 0. Note that this example

could be easily extended by considering a stochastic specification of σ to allow formore realistic classes of forward rate volatilities.

Case (i): Assumption 1 is obviously fulfilled and Assumption 2 follows for 0 ≤ s ≤ tfrom

1√t

∣∣∣Σ(s, t)ij

∣∣∣ =|σij |√t

∫ t

s

e−β(u−s) du ≤ 2√t− sβ√t|σij | ≤

2

β|σij | =: wij(s) .

Then, by Lemma 5.1 we get for the forward rate

f(t, T )(5.2)= f(0, T ) + 4

∫ t

0

Tr [σ(s, T )Xs (−Σ(s, T ))] ds+ 2

∫ t

0

Tr[σ(s, T )

√Xs dWs

](3.3)= f(0, T ) +

4

β

∫ t

0

(e−β(T−s) − e−2β(T−s)

)Tr[σ2Xs

]ds

+ 2

∫ t

0

e−β(T−s) Tr[σ√Xs dWs

](5.1)=

(A.3)f(0, T ) +

4

β

∫ t

0

(e−β(T−s) − e−2β(T−s)

)Tr[σ2Xs

]ds

− β∫ t

0

e−β(T−s) Tr[σXs] ds+ e−β(T−t) Tr[σXt]

− e−β T Tr[σX0]− a

βTr[σ]

(e−β(T−t) − e−β T

).

If we put

h0(t, T ) := f(0, T )− e−βT Tr[σX0]− a

βTr[σ]

(e−β(T−t) − e−βT

),


h(t, T ) :=

− 4βσ

2e−2βT

−βσe−βT + 4βσ

2e−βT

e−β(T−t)σ

,

and

Zt :=

∫ t0 e2βsXs ds∫ t0eβsXs dsXt

, (5.4)

we get

f(t, T ) = h0(t, T ) + h(t, T ) · Zt = h0(t, T ) + Tr[Z>t h(t, T )

].

Note that in this setting we obtain an affine d-dimensional realisation for the forwardrate as in Definition 3 of [11] with h0(t, T ) and h(t, T ) being Ft-measurable processesand Z an affine process.

Furthermore, the short rate process (rt)t≥0 has the form

rt(3.12)

= r0 − Tr[σX0] +

∫ t

0

φ(u) du+ Tr[σXt] .

with φ as in (3.13), since∫ t

0Tr[σ(u, u) dXu] = Tr[σ (Xt −X0)] for all t ≥ 0 by (5.3).

By Theorem 3.1 we get that

rt(3.10)

= f(0, t) +1

β

(1− β − e−βt

)Tr[σX0]− a

β

(1− e−βt

)Tr[σ] + Tr[σXt]

− β∫ t

0

e−β(t−s) Tr[σXs] ds−4

β

∫ t

0

(e−2β(t−s) − e−β(t−s)

)Tr[σ2Xs

]ds ,

hence the short-rate process r is a Markov process with respect to Z, defined in(5.4).The yield has the following form

Y (t, T )(3.16)

= Y (0; t, T ) +2

(T − t)β2

∫ t

0

((e−β(T−s) − 1

)2

−(e−β(t−s) − 1

)2)

Tr[σ2Xs

]ds

− 1

T−t

(∫ t

0

(e−β(T−s) − e−β(t−s)

)Tr[σXs] ds+

1

β

(1− e−β(T−t)

)Tr[σXt]

)− 1

(T−t)β

((e−βT −e−βt

)Tr[σX0] +

aTr[σ]

β

(e−β(T−t) −eβT +e−βt −1

)).

Then, the long-term drift for t ≥ 0 is

µ∞(t)(4.3)= lim

T→∞

Γ(t, T )

T

(3.18)= lim

T→∞

1

β2T

(e−β(T−t) − 1

)2

σXt σ = 0 , (5.5)

and we get by Theorem 4.1 that

`t(4.2)= `0 + 2

∫ t

0

Tr[µ∞(s)] ds(5.5)= `0 ,

i.e. (`t)t≥0 is constant, as expected since σ(t, T ) ∼ O(

1T−t

).

Case (ii): Assumption 1 is again fulfilled obviously and Assumption 2 follows for0 ≤ s ≤ t since

1√t

∣∣∣Σ(s, t)ij

∣∣∣ = 2 |σij |√t− s√t

= 2 |σij |√

1− s

t≤ 2 |σij | =: wij(s) .


The forward rate for 0 ≤ t ≤ T is

f(t, T )(5.2)= f(0, T ) + 4

∫ t

0

Tr [σ(s, T )Xs (−Σ(s, T ))] ds+ 2

∫ t

0

Tr[σ(s, T )

√Xs dWs

](3.3)= f(0, T ) + 8

∫ t

0

Tr[σ2Xs

]ds− 1

2

∫ t

0

(T−s)−32 Tr[σXs] ds

+1√T−t

Tr[σXt]−1√T

Tr[σX0]− 2aTr[σ](√

T −√T−t

)and the yield for [t, T ] has the following forms

Y (t, T )(3.16)

= Y (0; t, T ) + 8

∫ t

0

Tr[σ2 Xs

]ds+

1

T−t

∫ t

0

(1√T−s

− 1√t−s

)Tr[σXs] ds

+2 Tr[σXt]√

T−t−

2(√

T−√t)

Tr[σX0]

T−t +4

3aTr[σ]

(√T−t− T

√T − t

√t

T−t

).

Then, the long-term drift for t ≥ 0 is

µ∞(t)(4.3)= lim

T→∞

Γ(t, T )

T

(3.18)= lim

T→∞

4 (T − t)σXtσ

T= 4σXt σ , (5.6)

and we get with Theorem 4.1 that

`t(4.2)= `0 + 2

∫ t

0

Tr[µ∞(s)] ds(5.6)= `0 + 8

∫ t

0

Tr[σXs σ] ds .

Appendix A

Here we provide the proofs of Proposition 3.1 and Theorem 3.1. The resultsfollow by applying the Fubini theorem for integrable functions (cf. Theorem 14.16in Chapter 14 of [42]) and the stochastic Fubini theorem (cf. Theorem 65 in ChapterIV of [49]). For further details on the following computations, we refer to [34].

Proof of Proposition 3.1. Let us introduce for every maturity T > 0 the quantity

Z(t, T ) := −∫ T

t

f(t, u) du , (A.1)

for all 0 ≤ t ≤ T . From the dynamics of the forward rate (3.1) we deduce that forall T > 0

Z(t, T )(A.1)= −

∫ T

t

f(0, u) du−∫ T

t

∫ t

0

α(s, u) ds du−∫ T

t

∫ t

0

Tr[σ(s, u) dXs] du ,

(A.2)for all 0 ≤ t ≤ T . By combining (2.11), (3.3), (A.2), and (3.13), we derive thefollowing identity

Z(t, T )(A.2)= Z(0, T )+

∫ t

0

f(0, u) du−∫ T

t

∫ t

0

α(s, u) ds du−∫ T

t

∫ t

0

Tr[σ(s, u) dXs] du

(3.13)= Z(0, T ) +

∫ t

0

rs ds−∫ t

0

∫ T

s

α(s, u) du ds−∫ t

0

∫ T

s

Tr[σ(s, u) du dXs]

(2.11)=

(3.3)Z(0, T ) +

∫ t

0

rs ds−∫ t

0

∫ T

s

α(s, u) du ds

+

∫ t

0

Tr[Σ(s, T )

(√Xs dWsQ+Q>dW>s

√Xs

)]+

∫ t

0

Tr[Σ(s, T ) (b+B(Xs))] ds+

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]µX(ds, dξ) .


Note that in general for A,B ∈Md with A symmetric, i.e. A ∈ Sd, it holds

Tr[A(B +B>

)]= Tr[AB] + Tr

[AB>

]= Tr[AB] + Tr

[(BA)

>]

= Tr[AB] + Tr [BA] = 2 Tr[AB] . (A.3)

Therefore, we get due to σ(s, t) ∈ Sd for all s, t ≥ 0 that

Z(t, T )(A.3)= Z(0, T )+

∫ t

0

rs ds−∫ t

0

∫ T

s

α(s, u) du ds+2

∫ t

0

Tr[Σ(s, T )

√Xs dWsQ

]+

∫ t

0

Tr[Σ(s, T ) (b+B(Xs))] ds+

∫ t

0

∫S+d\0

Tr[Σ(s, T ) ξ]µX(ds, dξ) .

(A.4)

Note that for all 0 ≤ t ≤ T

∆Z(t, T ) = Tr [Σ(t, T ) ∆Xt] . (A.5)

With the help of (A.4) and the fact that

〈Wlm,Wru〉s =

s if l = r and m = u,0 else,

(A.6)

we can calculate the quadratic variation of Z for all T > 0 as follows

〈Z( · , T )〉ct =

⟨Tr

[∫ ·0

Σ(s, T )√Xs dWsQ

]⟩t

(A.6)= 4

∫ t

0

Tr[QΣ(s, T )Xs Σ(s, T )Q>

]ds .

(A.7)

Further, we see that due to equation (2.27) of [13] for all u ∈ S+d and a process Y

on S+d it holds

Tr[B>(u)Y

]= Tr[B(Y )u] , (A.8)

where B is defined according to (2.13) and therefore for all 0 ≤ t ≤ T and α = Q>Q∫ t

0

Tr[Σ(s, T ) (b+B(Xs))] + 2 Tr[QΣ(s, T )Xs Σ(s, T )Q>

]ds

+

∫ t

0

∫S+d\0

(eTr[Σ(s,T ) ξ] − 1

)ν(ds, dξ)

(2.12)=

(A.8)−∫ t

0

Tr[−Σ(s, T ) b− 2 Σ(s, T )αΣ(s, T )Xs +B>(−Σ(s, T ))Xs

]ds

+

∫ t

0

∫S+d\0

(e−Tr[−Σ(s,T ) ξ] − 1

)m(dξ) ds

+

∫ t

0

Tr

[Xs

∫S+d\0

(e−Tr[−Σ(s,T ) ξ] − 1

)µ(dξ)

]ds

(2.9)=

(2.8)

∫ t

0

(−F (−Σ(s, T ))− Tr[R(−Σ(s, T ))Xs]) ds . (A.9)

Now, we apply Ito’s formula on P (t, T ) := exp(Z(t, T )) for every maturity T > 0(cf. Definition 1.4.2 of [8]) and obtain the following representation, where we useProposition 1.28 of Chapter II in [38] to combine the measures µX(ds, dξ) and


ν(ds, dξ), since the affine process X has only jumps of finite variation (cf. (2.10)).

P (t, T ) = P (0, T ) +

∫ t

0

P (s−, T ) dZ(s, T ) +1

2

∫ t

0

P (s, T ) d 〈Z( · , T )〉cs

+

∆Z(s,T )6=0∑0<s≤t

[eZ(s,T ) − eZ(s−,T ) −∆Z(s, T ) eZ(s−,T )

](A.5)=

(A.7)P (0, T ) +

∫ t

0

P (s−, T ) dZ(s, T )

+ 2

∫ t

0

P (s, T ) Tr[QΣ(s, T )Xs Σ(s, T )Q>

]ds

+

∆Xs 6=0∑0≤s≤t

[eZ(s,T ) − eZ(s−,T ) − Tr[Σ(s, T ) ∆Xs] e

Z(s−,T )]

(A.4)=

(2.1)P (0, T ) + 2

∫ t

0

P (s, T ) Tr[Σ(s, T )

√Xs dWsQ

]+

∫ t

0

P (s, T )

(rs −

∫ T

s

α(s, u) du

)ds

+

∫ t

0

P (s, T ) Tr[Σ(s, T ) (b+B(Xs))] ds

+

∫ t

0

P (s−, T )

∫S+d\0

Tr[Σ(s, T ) ξ] µX(ds, dξ)

+ 2

∫ t

0

P (s, T ) Tr[QΣ(s, T )Xs Σ(s, T )Q>

]ds

+

∆Xs 6=0∑0≤s≤t

[e∆Z(s,T )P (s−, T )−P (s−, T )−Tr[Σ(s, T ) ∆Xs]P (s−, T )

](A.5)=

(A.9)P (0, T ) +

∫ t

0

P (s−, T ) (rs +A(s, T )) ds+ 2

∫ t

0

P (s, T ) Tr[Σ(s, T )

√Xs dWsQ

]+

∫ t

0

P (s−, T )

∫S+d\0

(eTr[Σ(s,T ) ξ] − 1

)(µX − ν

)(ds, dξ) .

Assumption 1 guarantees that all integrals above are finite.

Proof of Theorem 3.1. By using (3.7) we see that the discounted bond priceprocess under Q is

P (t, T )

βt

(3.8)= P (0, T ) +

∫ t

0

P (s, T )

βs

(A(s, T ) + 2 Tr

[Σ(s, T )

√Xs γsQ

])ds

+ 2

∫ t

0

P (s, T )

βsTr[Σ(s, T )

√Xs dW

∗s Q]

+

∫ t

0

∫S+d\0

P (s−, T )

βs

(eTr[Σ(s,T ) ξ]−1

) (µX− ν∗

)(ds, dξ)

+

∫ t

0

∫S+d\0

P (s−, T )

βs


)(K(s, ξ)−1) ν(ds, dξ)

(A.10)


for all 0 ≤ t ≤ T . Since P (t,T )βt

, t ≤ T , has to be a local martingale under Q, the

drift in (A.10) must disappear, i.e. for all 0 ≤ t ≤ T

0(2.12)

=

∫ t

0

P (s, T )

βsA(s, T ) ds+ 2

∫ t

0

P (s, T )

βsTr[Σ(s, T )

√Xs γsQ

]ds

+

∫ t

0

∫S+d\0

P (s−, T )

βs


)(K(s, ξ)−1) (m(dξ)+Tr[Xsµ(dξ)]) ds.

It follows for all 0 ≤ t ≤ T that

A(t, T ) = −2 Tr[Σ(t, T )

√Xt γtQ

]−∫S+d\0

(eTr[Σ(t,T ) ξ]−1

)(K(t, ξ)−1) (m(dξ)+Tr[Xtµ(dξ)])

dt⊗ dP-a.s.Consequently we get for all 0 ≤ t ≤ T by Satz 6.28 in [42]

α(t, T )(3.4)= −∂TA(t, T )− ∂TF (−Σ(t, T ))− ∂T Tr[R(−Σ(t, T ))Xt]

(A.3)= −Tr

[σ(t, T )

(b+B(Xt) + 2

√Xt γtQ

)]−4 Tr

[Qσ(t, T )Xt Σ(t, T )Q>

]−∫S+d\0

Tr [σ(t, T ) ξ] eTr[Σ(t,T ) ξ]K(t, ξ) (m(dξ)+Tr[Xtµ(dξ)])

dt⊗ dP-a.s.Hence, P (t,T )

βt, t ≤ T , is a Q-local martingale if and only if equation (3.10) is

fulfilled dt⊗ dP-a.s. Equation (3.10) represents the HJM condition on the drift inthe affine setting on S+

d . Then, the forward rate under Q follows a process of theform

f(t, T )(3.1)=

(2.11)f(0, T ) +

∫ t

0

α(s, T ) ds+

∫ t

0

Tr[σ(s, T )

(b+B(Xs) + 2

√Xs γsQ

)]ds

+

∫ t

0

∫S+d\0

Tr[σ(s, T ) ξ] µX(ds, dξ) + 2

∫ t

0

Tr[σ(s, T )

√Xs dW

∗s Q]

(3.10)= f(0, T )− 4

∫ t

0

Tr[Qσ(s, T ) Xs Σ(s, T ) Q>

]ds

−∫ t

0

∫S+d\0

Tr[σ(s, T ) ξ] eTr[Σ(s,T ) ξ]K(s, ξ) ν(ds, dξ)

+

∫ t

0

∫S+d\0

Tr[σ(s, T ) ξ] µX(ds, dξ) + 2

∫ t

0

Tr[σ(s, T )

√Xs dW

∗s Q]

(3.3)= f(0, T ) + 4

∫ t

0

Tr

[Qσ(s, T )Xs

∫ T

s

σ(s, u) du Q>]ds

+

∫ t

0

∫S+d\0


)(ds, dξ)

−∫ t

0

∫S+d\0

Tr[σ(s, T ) ξ](eTr[Σ(s,T ) ξ] − 1

)ν∗(ds, dξ)

+ 2

∫ t

0

Tr[σ(s, T )

√Xs dW

∗s Q]

(2.12)= f(0, T ) +

∫ t

0

4 Tr

[Qσ(s, T )Xs

∫ T

s

σ(s, u) du Q>]

−∫S+d\0

K(s, ξ) Tr[σ(s, T ) ξ](eTr[Σ(s,T ) ξ]−1

)(m(dξ)+Tr[Xsµ(dξ)])

ds


+

∫ t

0

∫S+d\0


)(ds, dξ)

+ 2

∫ t

0

Tr[σ(s, T )

√Xs dW

∗s Q],

where we have used again Proposition 1.28 of Chapter II in [38].

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(F. Biagini) Department of Mathematics, LMU University

Theresienstrasse 39, D-80333 Munich, Germany.

Secondary Affiliation: Department of Mathematics, University of Oslo, Norway.E-mail address: [email protected]

URL: http://www.mathematik.uni-muenchen.de/personen/professoren/biagini/index.html

(A. Gnoatto) Department of Mathematics, LMU University

Theresienstrasse 39, D-80333 Munich, Germany

E-mail address: [email protected]

URL: http://www.alessandrognoatto.com

(M. Hartel) MEAG Munich ERGO AssetManagement GmbHOskar-von-Miller-Ring 18, D-80333 Munich, Germany

E-mail address: [email protected]

URL: http://www.mathematik.uni-muenchen.de/personen/mitarbeiter/haertel/index.html

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